We consider the teleportation protocol introduced in [1]}. Here, we are considering the parties involved in the teleportation are a ground station and a satellite in space, with the quantum channel between them corresponding to the free-space atmospheric channel as exemplified in Fig. REF a). The teleportation protocol starts with the generation of a bipartite entangled resource state, \({\Xi }_{AB}\) , in the satellite. Part \(A\)  of \({\Xi }_{AB}\)  is sent through the atmosphere to the ground station, where it is combined with the input state using a balanced beam-splitter. Afterwards, a Bell projective measurement (using a pair of homodyne detectors) on part \(A\)  and the input state is performed. The measurement result is broadcast to the satellite which, by doing a corrective operation on \(B\) , recovers the input state as the final output of the protocol.We assume noiseless classical communications between satellite and ground station by means of a different channel, such as radio-waves or wide-beam optical signals.
To describe the teleportation protocol we follow the methodology introduced
in [2]}. Using this methodology the output state can be computed by using the Wigner characteristic functions (CF) of the input state \({\Xi }_\mathrm {in}\)  and the entangled resource state \({\Xi }_{AB}\) . Here, we indicate CFs by \(\makebox{\large $\chi $}(\xi )\) , for some complex parameter \(\xi \) .
In [3]} the methodology is further expanded to include imperfect homodyne measurements, obtaining
\(\makebox{\large $\chi $}_{\mathrm {out}}(\xi ) =& \makebox{\large $\chi $}_{\mathrm {in}}(g \eta \xi ) \nonumber \\&\times \makebox{\large $\chi $}_{AB}(\xi , g \eta \xi ^*) e^{-\frac{|\xi |^2}{2} g^2 (1 - \eta ^2) },\) 
